Question

1. The life insurance industry maintains that the average worker in Saskatoon has no more than $25,000 of personal life insurance. You believe this to be too low. You sample 100 workers in Saskatoon at random and find the sample average to be $26,650 of personal life insurance. The population standard deviation is known to be $12,000. Use α=0.05 throughout.

1. Test your belief using a significance level of 5%.
2. Explain, in the context of this question, what is meant by a Type I error, a Type II error, and the power of the test?
3. If the true average for this population is in fact $30,000, what is the probability of committing a Type II error?

(d) Calculate the power of the test

Answer 1

a)

Let the personal life insurance of a worker be given by X which is distributed with mean and standard dev. .

We test the hypotheses

Let the sample mean be . Since n = 100, std. dev. of = .

We reject is s.t. under the null,

Assuming Normal distribution of sample mean, that value is

Since our sampled mean is only 26650 we fail to reject the null hypothesis at a 5% significance level.

b)

Type 1 error is when we reject the claim of the company, when it is in fact, true.

Type 2 error is when we the claim of is actually false, but we don't have enough evidence to reject it.

Power of the test is Probability of having enough evidence to reject the claim, when it is false.

c) We reject when sample mean exceeds c.

assuming the Normal approximation.

d) Power of the test is

At , this is 0.95